Relative motion

The state of motion of a body depends on the reference system with respect to which we are describing it. Relative motion is the part of Kinematics that deals with finding relationships (equations) between the position, velocity and acceleration vectors that different observers measure.

In these pages we are going to solve problems of relative motion in two different situations:

  • Constant relative motion: One of the reference frames (O) is at rest and the other one (O’) has a constant velocity V with respect to the first one. Both systems of reference are inertial reference frames because they don’t have acceleration and they don’t rotate.
  • Relative motion with rotating axes: One of the reference frames (O) is at rest (inertial frame) and the other (O’) rotates with constant angular velocity ω with respect to the first. The second system of reference is a non-inertial reference frame.

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Constant relative motion: The equations that transform the position, velocity and acceleration vectors between by both observers are known as Galilean transformations.

In the figure below the two inertial frames of reference as well as the position vector that has a moving particle for each one of them are represented.

If we assume that at the initial time the origin of the two frames of reference coincides, the Galilean transformations are given by:

Where the variables without prime denote the vectors measured by O and the variables with prime are the corresponding vectors measured by O’.

The first equation can be deduced geometrically from the figure. Since the initial position of both origins is the same, at a time t O’ will have traveled a distance Vt with respect to O.

The relationship between velocities is deduced by deriveng the relationship between the position vectors. And the equation that relates the acceleration measured by both observers is obtained by deriving the equation that relates the velocities.

As you can see, the acceleration vector is the same for all the inertial frames of reference. Therefore, the laws of Physics are the same in all inertial reference frames.

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Relative motion with rotating axes: Below are the equations that relate the position, velocity and acceleration vectors. As you can see in the figure below, we will assume that the origin of both systems of reference are in the same position.

As you can see, the acceleration of the particle measured by the non inertial frame of reference is different from that measured by the inertial one.

The terms that appear in the equation relating both acceleration vectors are respectively the Coriolis acceleration and the centrifugal acceleration.

Both terms are responsible for the so called inertial forces. A non inertial frame of reference using Newton’s laws to study the motion of the particle “interprets” that these accelerations have to be produced by a force. But there isn’t any real physical interaction that produces them. There are the result of the observer own motion. This is why the inertial forces are also called fictitious forces o pseudo forces.

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